cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A224731 b(n+1) - b(n) + n where b(n) = A095114(n).

Original entry on oeis.org

2, 4, 5, 7, 8, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75
Offset: 1

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Author

Robert Price, Apr 16 2013

Keywords

Comments

This sequence, requested in A095114, is a companion to A109512.
Every positive integer is either of the form a(n)+n-1 (A109512) or of the form a(n+1)-a(n)+n (A224731), but not both.

Links

Crossrefs

Cf. A095114.

A001462 Golomb's sequence: a(n) is the number of times n occurs, starting with a(1) = 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

It is understood that a(n) is taken to be the smallest number >= a(n-1) which is compatible with the description.
In other words, this is the lexicographically earliest nondecreasing sequence of positive numbers which is equal to its RUNS transform. - N. J. A. Sloane, Nov 07 2018
Also called Silverman's sequence.
Vardi gives several identities satisfied by A001463 and this sequence.
We can interpret this sequence as a triangle: start with 1; 2,2; 3,3; and proceed by letting the row sum of row m-1 be the number of elements of row m. The partial sums of the row sums give 1, 5, 11, 38, 272, ... Conjecture: this proceeds as Lionel Levine's sequence A014644. See also A113676. - Floor van Lamoen, Nov 06 2005
A Golomb-type sequence, that is, one with the property of being a sequence of run length of itself, can be built over any sequence with distinct terms by repeating each term a corresponding number of times, in the same manner as a(n) is built over natural numbers. See cross-references for more examples. - Ivan Neretin, Mar 29 2015
From Amiram Eldar, Jun 19 2021: (Start)
Named after the American mathematician Solomon Wolf Golomb (1932-2016).
Guy (2004) called it "Golomb's self-histogramming sequence", while in previous editions of his book (1981 and 1994) he called it "Silverman's sequence" after David Silverman. (End)
a(n) is also the number of numbers that occur n times. - Leo Crabbe, Feb 15 2025

Examples

			a(1) = 1, so 1 only appears once. The next term is therefore 2, which means 2 appears twice and so a(3) is also 2 but a(4) must be 3. And so on.
G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 4*x^8 + ... - _Michael Somos_, Nov 07 2018

References

  • Graham Everest, Alf van der Poorten, Igor Shparlinski and Thomas Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 10.
  • Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 66.
  • Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section E25, p. 347-348.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001463 (partial sums) and A262986 (start of first run of length n).
First differences are A088517.
Golomb-type sequences over various substrates (from Glen Whitney, Oct 12 2015):
A000002 and references therein (over periodic sequences),
A109167 (over nonnegative integers),
A080605 (over odd numbers),
A080606 (over even numbers),
A080607 (over multiples of 3),
A169682 (over primes),
A013189 (over squares),
A013322 (over triangular numbers),
A250983 (over integral sums of itself).
Applying "ee Rabot" to this sequence gives A319434.
See also A095114.

Programs

  • Haskell
    a001462 n = a001462_list !! (n-1)
a001462_list = 1 : 2 : 2 : g 3  where
   g x = (replicate (a001462 x) x) ++ g (x + 1)
-- Reinhard Zumkeller, Feb 09 2012
  • Magma
    [ n eq 1 select 1 else 1+Self(n-Self(Self(n-1))) : n in [1..100] ]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Maple
    N:= 10000: A001462[1]:= 1: B[1]:= 1: A001462[2]:= 2:
for n from 2 while B[n-1] <= N do
  B[n]:= B[n-1] + A001462[n];
  for j from B[n-1]+1 to B[n] do A001462[j]:= n end do
end do:
seq(A001462[j],j=1..N); # Robert Israel, Oct 30 2012
  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; Table[ a[n], {n, 84}] (* Robert G. Wilson v, Aug 26 2005 *)
GolSeq[n_]:=Nest[(k = 0; Flatten[# /. m_Integer :> (ConstantArray[++k,m])]) &, {1, 2}, n]
GolList=Nest[(k = 0;Flatten[# /.m_Integer :> (ConstantArray[++k,m])]) &, {1, 2},7]; AGolList=Accumulate[GolList]; Golomb[n_]:=Which[ n <= Length[GolList], GolList[[n]], n <= Total[GolList],First[FirstPosition[AGolList, ?(# > n &)]], True, $Failed] (* _JungHwan Min, Nov 29 2015 *)
  • PARI
    a = [1, 2, 2]; for(n=3, 20, for(i=1, a[n], a = concat(a, n))); a /* Michael Somos, Jul 16 1999 */
  • PARI
    {a(n) = my(A, t, i); if( n<3, max(0, n), A = vector(n); t = A[i=2] = 2; for(k=3, n, A[k] = A[k-1] + if( t--==0, t = A[i++]; 1)); A[n])}; /* Michael Somos, Oct 21 2006 */
  • Python
    a=[0, 1, 2, 2]
for n in range(3, 21):a+=[n for i in range(1, a[n] + 1)]
a[1:] # Indranil Ghosh, Jul 05 2017

Formula

a(n) = phi^(2-phi)*n^(phi-1) + E(n), where phi is the golden number (1+sqrt(5))/2 (Marcus and Fine) and E(n) is an error term which Vardi shows is O( n^(phi-1) / log n ).
a(1) = 1; a(n+1) = 1 + a(n+1-a(a(n))). - Colin Mallows
a(1)=1, a(2)=2 and for a(1) + a(2) + ... + a(n-1) < k <= a(1) + a(2) + ... + a(n) we have a(k)=n. - Benoit Cloitre, Oct 07 2003
G.f.: Sum_{n>0} a(n) x^n = Sum_{k>0} x^a(k). - Michael Somos, Oct 21 2006
a(A095114(n)) = n and a(m) < n for m < A095114(n). - Reinhard Zumkeller, Feb 09 2012 [First inequality corrected from a(m) < m by Glen Whitney, Oct 06 2015]
Conjecture: a(n) >= n^(phi-1) for all n. - Jianing Song, Aug 19 2021
a(n) = A095114(n+1) - A095114(n). - Alan Michael Gómez Calderón, Dec 21 2024 after Ralf Stephan

A129299 a(1)=1, a(n) = a(n-1) + (sum of the earlier terms of the sequence which are <= n).

Original entry on oeis.org

1, 2, 5, 8, 16, 24, 32, 48, 64, 80, 96, 112, 128, 144, 160, 192, 224, 256, 288, 320, 352, 384, 416, 472, 528, 584, 640, 696, 752, 808, 864, 952, 1040, 1128, 1216, 1304, 1392, 1480, 1568, 1656, 1744, 1832, 1920, 2008, 2096, 2184, 2272, 2408, 2544, 2680, 2816
Offset: 1

Views

Author

Leroy Quet, Apr 08 2007

Keywords

Examples

			The terms that are <= 9 are a(1) through a(4). So a(9) = a(8) + a(1) + a(2) + a(3) + a(4) = 48 + 1 + 2 + 5 + 8 = 64.

Links

Crossrefs

Cf. A129300, A126022, A095114.

Programs

  • Haskell
    a129299 n = a129299_list !! (n-1)
a129299_list = 1 : f [1] 2 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + sum [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012
  • Maple
    a[1]:=1: for n from 2 to 60 do b:=a[n-1]: for j from 1 to n-1 do if a[j]<=n then b:=b+a[j] else b:=b: fi: od: a[n]:=b: od: seq(a[n],n=1..60); # Emeric Deutsch, Apr 10 2007
  • Mathematica
    s={1};Do[AppendTo[s,s[[-1]]+Total[Select[s,#<=n&]]],{n,2,51}];s (* James C. McMahon, Jan 20 2025 *)

Extensions

More terms from Emeric Deutsch, Apr 10 2007

A129300 a(0)=1. a(n) = a(n-1) + (sum of the terms of the sequence which are <= n).

Original entry on oeis.org

1, 2, 5, 8, 11, 19, 27, 35, 51, 67, 83, 110, 137, 164, 191, 218, 245, 272, 299, 345, 391, 437, 483, 529, 575, 621, 667, 740, 813, 886, 959, 1032, 1105, 1178, 1251, 1359, 1467, 1575, 1683, 1791, 1899, 2007, 2115, 2223, 2331, 2439, 2547, 2655, 2763, 2871, 2979
Offset: 0

Views

Author

Leroy Quet, Apr 08 2007

Keywords

Examples

			The terms that are <= 9 are a(0) through a(3). So a(9) = a(8) + a(0) + a(1) + a(2) + a(3) = 51 + 1 + 2 + 5 + 8 = 67.

Links

Crossrefs

Cf. A129299, A126022, A095114.

Programs

  • Haskell
    a129300 n = a129300_list !! (n-1)
a129300_list = 1 : f [1] 1 where
   f xs@(x:_) k = y : f (y:xs) (k+1) where
     y = x + sum [z | z <- xs, z <= k]
-- Reinhard Zumkeller, Feb 09 2012
  • Maple
    a[0]:=1: for n from 1 to 60 do b:=0: for j from 0 to n-1 do if a[j]<=n then b:=b+a[j] else fi od: a[n]:=a[n-1]+b: od: seq(a[n],n=0..60); # Emeric Deutsch, Apr 12 2007
  • Mathematica
    Nest[Function[{s,n},Append[s,s[[-1]]+Total[TakeWhile[s,#<=n&]]]]@@{#,Length[#]}&,{1}, 50] (* James C. McMahon, Jan 20 2025 *)

Extensions

More terms from Emeric Deutsch, Apr 12 2007

A262986 Start of first run of length n in Golomb's sequence A001462.

Original entry on oeis.org

1, 2, 6, 12, 24, 39, 63, 91, 123, 168, 218, 273, 345, 423, 507, 597, 709, 828, 954, 1087, 1247, 1415, 1591, 1775, 1991, 2216, 2450, 2693, 2945, 3235, 3535, 3845, 4165, 4495, 4869, 5254, 5650, 6057, 6475, 6943, 7423, 7915, 8419, 8935, 9463, 10048, 10646, 11257, 11881, 12518
Offset: 1

Views

Author

Glen Whitney, Oct 06 2015

Keywords

Comments

The first run of length n in A001462 begins with A095114(n), i.e., A001462(a(n)) = A095114(n).
a(n) = A095114(A095114(n)).

Examples

			Since Golomb's sequence starts 1,2,2,3,3,4,4,4,5,... the first run of length three starts at position 6, so a(3) = 6.

Crossrefs

Cf. A001462, A095114.

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1 + a[n - a[a[n - 1]]]; s = Array[a, {13000}]; t = Most@ Map[Last, Tally@ s]; Table[Total@ Take[t, First@ Flatten@ Position[t, n]] - n + 1, {n, 50}] (* Michael De Vlieger, Oct 07 2015, after Robert G. Wilson v at A001462 *)
  • PARI
    a(n) = if (n<2, 1, a(n-1)+ t(n-1)*(n-1));
t(n) = local(A, t, i); if(n<3, max(0, n), A=vector(n); t=A[i=2]=2; for(k=3, n, A[k]=A[k-1]+if(t--==0, t=A[i++ ]; 1)); A[n]);
vector(100, n, a(n))\\ Altug Alkan, Oct 06 2015

Formula

a(1) = 1; a(n+1) = a(n) + A001462(n)*n.

Extensions

More terms from Altug Alkan, Oct 06 2015
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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